Solution of inequalities with one variable and their systems. Systems of inequalities - initial information

The topic of the lesson is "Solving inequalities and their systems" (mathematics grade 9)

Lesson type: lesson of systematization and generalization of knowledge and skills

Lesson technology: critical thinking development technology, differentiated learning, ICT technologies

The purpose of the lesson: repeat and systematize knowledge about the properties of inequalities and methods for solving them, create conditions for the formation of skills to apply this knowledge in solving standard and creative problems.

Tasks.

Educational:

to promote the development of students' skills to summarize the knowledge gained, to analyze, synthesize, compare, draw the necessary conclusions

organize the activities of students to apply the acquired knowledge in practice

to promote the development of skills to apply the acquired knowledge in non-standard conditions

Developing:

continue the formation of logical thinking, attention and memory;

improve the skills of analysis, systematization, generalization;

creating conditions that ensure the formation of self-control skills in students;

promote the acquisition of the necessary skills for independent learning activities.

Educational:

to cultivate discipline and composure, responsibility, independence, a critical attitude towards oneself, attentiveness.

Planned educational outcomes.

Personal: responsible attitude to learning and communicative competence in communication and cooperation with peers in the process of educational activities.

Cognitive: the ability to define concepts, create generalizations, independently choose the grounds and criteria for classification, build logical reasoning, draw conclusions;

Regulatory: the ability to identify potential difficulties in solving an educational and cognitive task and find means to eliminate them, to evaluate their achievements

Communicative: the ability to express judgments using mathematical terms and concepts, formulate questions and answers in the course of the assignment, share knowledge between group members to make effective joint decisions.

Basic terms, concepts: linear inequality, quadratic inequality, system of inequalities.

Equipment

Projector, teacher's laptop, several netbooks for students;

Presentation;

Cards with basic knowledge and skills on the topic of the lesson (Appendix 1);

Cards with independent work (Appendix 2).

Lesson plan

List of used literature: Algebra. Textbook for grade 9. / Yu.N.Makrychev, N.G.Mindyuk, K.I.Neshkov, S.B.Suvorova. - M.: Enlightenment, 2014

Lesson topic: Solving a system of linear inequalities with one variable

The date: _______________

Class: 6a, 6b, 6c

Lesson type: learning new material and primary consolidation.

Didactic goal: create conditions for understanding and comprehending a block of new educational information.

Objectives: 1) Educational: introduce concepts: solution of systems of inequalities, equivalent systems of inequalities and their properties; teach how to apply these concepts when solving the simplest systems of inequalities with one variable.

2) Developing: to promote the development of elements of creative, independent activity of students; develop speech, the ability to think, analyze, summarize, express one's thoughts clearly, concisely.

3) Educational: fostering a respectful attitude towards each other and a responsible attitude to educational work.

Tasks:

repeat the theory on the topic of numerical inequalities and numerical gaps;

give an example of a problem that is solved by a system of inequalities;

consider examples of solving systems of inequalities;

do independent work.

Forms of organization of educational activities:- frontal - collective - individual.

Methods: explanatory - illustrative.

Lesson plan:

1. Organizational moment, motivation, goal setting

2. Updating the study of the topic

3. Learning new material

4. Primary fixation and application of new material

5. Doing your own work

7. Summing up the lesson. Reflection.

During the classes:

1. Organizational moment

Inequality can be a good helper. You just need to know when to call for help. The language of inequalities is often used to formulate problems in many applications of mathematics. For example, many economic problems are reduced to the study of systems of linear inequalities. Therefore, it is important to be able to solve systems of inequalities. What does it mean to “solve the system of inequalities”? That's what we'll cover in today's lesson.

2. Actualization of knowledge.

oral work with the class three students work on individual cards.

To repeat the theory of the topic "Inequalities and their properties", we will conduct testing, followed by a test and a conversation on the theory of this topic. Each test task involves the answer "Yes" - a figure, "No" - a figure ____

As a result of the test, some figure should be obtained.

(answer: ).

Establish a correspondence between inequality and a numerical gap

1. (– ; – 0,3)

2. (3; 18)

3. [ 12; + )

4. (– 4; 0]

5. [ 4; 12]

6. [ 2,5; 10)

"Mathematics teaches us to overcome difficulties and correct our own mistakes." Find an error in the solution of the inequality, explain why the error was made, write down the correct solution in your notebook.

2x<8-6

x>-1

3. Learning new material.

What do you think is called the solution of a system of inequalities?

(The solution of a system of inequalities with one variable is the value of the variable for which each of the inequalities of the system is true)

What does "Solve a system of inequalities" mean?

(To solve a system of inequalities means to find all its solutions or to prove that there are no solutions)

What should be done to answer the question "Is the given number

a solution to a system of inequalities?

(Substitute this number in both inequalities of the system, if true inequalities are obtained, then the given number is a solution to the system of inequalities, if incorrect inequalities are obtained, then the given number is not a solution to the system of inequalities)

Formulate an algorithm for solving systems of inequalities

1. Solve each inequality of the system.

2. Draw graphically the solutions of each inequality on the coordinate line.

3. Find the intersection of solutions of inequalities on the coordinate line.

4. Write down the answer as a numerical interval.

Consider examples:

Answer:

Answer: no solution

4. Fixing the topic.

Working with textbook No. 1016, No. 1018, No. 1022

5. Independent work by options (Cards-tasks for students on the tables)

Independent work

Option 1

Solve the system of inequalities:

This article has collected initial information about systems of inequalities. Here we give a definition of a system of inequalities and a definition of a solution to a system of inequalities. It also lists the main types of systems that you most often have to work with in algebra lessons at school, and examples are given.

Page navigation.

What is a system of inequalities?

It is convenient to define systems of inequalities in the same way as we introduced the definition of a system of equations, that is, according to the type of record and the meaning embedded in it.

Definition.

System of inequalities is a record representing a certain number of inequalities written one below the other, united on the left by a curly bracket, and denoting the set of all solutions that are simultaneously solutions to each inequality of the system.

Let us give an example of a system of inequalities. Take two arbitrary , for example, 2 x−3>0 and 5−x≥4 x−11 , write them one under the other
2x−3>0 ,
5−x≥4 x−11
and unite with the sign of the system - a curly bracket, as a result we get a system of inequalities of the following form:

Similarly, an idea is given about systems of inequalities in school textbooks. It is worth noting that the definitions in them are given more narrowly: for inequalities with one variable or with two variables.

The main types of systems of inequalities

It is clear that there are infinitely many different systems of inequalities. In order not to get lost in this diversity, it is advisable to consider them in groups that have their own distinctive features. All systems of inequalities can be divided into groups according to the following criteria:

• by the number of inequalities in the system;
• by the number of variables involved in the recording;
• by the nature of the inequalities.

According to the number of inequalities included in the record, systems of two, three, four, etc. are distinguished. inequalities. In the previous paragraph, we gave an example of a system that is a system of two inequalities. Let us show another example of a system of four inequalities .

Separately, we say that it makes no sense to talk about a system of one inequality, in this case, in fact, we are talking about the inequality itself, and not about the system.

If you look at the number of variables, then there are systems of inequalities with one, two, three, etc. variables (or, as they say, unknowns). Look at the last system of inequalities written two paragraphs above. This is a system with three variables x , y and z . Note that her first two inequalities do not contain all three variables, but only one of them. In the context of this system, they should be understood as inequalities with three variables of the form x+0 y+0 z≥−2 and 0 x+y+0 z≤5, respectively. Note that the school focuses on inequalities with one variable.

It remains to discuss what types of inequalities are involved in writing systems. At school, they mainly consider systems of two inequalities (less often - three, even more rarely - four or more) with one or two variables, and the inequalities themselves are usually whole inequalities first or second degree (less often - higher degrees or fractionally rational). But do not be surprised if in the preparation materials for the OGE you come across systems of inequalities containing irrational, logarithmic, exponential and other inequalities. As an example, we present the system of inequalities , it is taken from .

What is the solution of a system of inequalities?

We introduce another definition related to systems of inequalities - the definition of a solution to a system of inequalities:

Definition.

Solving a system of inequalities with one variable such a value of a variable is called that turns each of the inequalities of the system into true, in other words, is the solution to each inequality of the system.

Let's explain with an example. Let's take a system of two inequalities with one variable . Let's take the value of the variable x equal to 8 , it is a solution to our system of inequalities by definition, since its substitution into the inequalities of the system gives two correct numerical inequalities 8>7 and 2−3 8≤0 . On the contrary, the unit is not a solution to the system, since when it is substituted for the variable x, the first inequality will turn into an incorrect numerical inequality 1>7 .

Similarly, we can introduce the definition of a solution to a system of inequalities with two, three, or more variables:

Definition.

Solving a system of inequalities with two, three, etc. variables called a pair, triple, etc. values ​​of these variables, which is simultaneously a solution to each inequality of the system, that is, it turns each inequality of the system into a true numerical inequality.

For example, a pair of values ​​x=1 , y=2 , or in another notation (1, 2) is a solution to a system of inequalities with two variables, since 1+2<7 и 1−2<0 - верные числовые неравенства. А пара (3,5, 3) не является решением этой системы, так как второе неравенство при этих значениях переменных дает неверное числовое неравенство 3,5−3<0 .

Systems of inequalities may have no solutions, may have a finite number of solutions, or may have infinitely many solutions. One often speaks of a set of solutions to a system of inequalities. When a system has no solutions, then there is an empty set of its solutions. When there are a finite number of solutions, then the set of solutions contains a finite number of elements, and when there are infinitely many solutions, then the set of solutions consists of an infinite number of elements.

Some sources introduce definitions of a particular and general solution to a system of inequalities, as, for example, in Mordkovich's textbooks. Under a particular solution to the system of inequalities understand its one single solution. In its turn general solution of the system of inequalities- these are all her private decisions. However, these terms make sense only when it is required to emphasize which solution is being discussed, but usually this is already clear from the context, so it is much more common to simply say "solution of a system of inequalities."

From the definitions of a system of inequalities and its solutions introduced in this article, it follows that the solution to a system of inequalities is the intersection of the sets of solutions of all inequalities of this system.

Bibliography.

1. Algebra: textbook for 8 cells. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M. : Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
2. Algebra: Grade 9: textbook. for general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M. : Education, 2009. - 271 p. : ill. - ISBN 978-5-09-021134-5.
3. Mordkovich A. G. Algebra. Grade 9 At 2 pm Part 1. Textbook for students of educational institutions / A. G. Mordkovich, P. V. Semenov. - 13th ed., Sr. - M.: Mnemosyne, 2011. - 222 p.: ill. ISBN 978-5-346-01752-3.
4. Mordkovich A. G. Algebra and beginning of mathematical analysis. Grade 11. At 2 hours. Part 1. A textbook for students of educational institutions (profile level) / A. G. Mordkovich, P. V. Semenov. - 2nd ed., erased. - M.: Mnemosyne, 2008. - 287 p.: ill. ISBN 978-5-346-01027-2.
5. USE-2013. Mathematics: typical examination options: 30 options / ed. A. L. Semenova, I. V. Yashchenko. - M .: Publishing house "National Education", 2012. - 192 p. - (USE-2013. FIPI - school).

1. The concept of inequality with one variable

2. Equivalent inequalities. Equivalence theorems for inequalities

3. Solving inequalities with one variable

4. Graphical solution of inequalities with one variable

5. Inequalities containing a variable under the modulus sign

6. Key Findings

Inequalities with one variable

Offers 2 X + 7 > 10's, x 2 +7x< 2,(х + 2)(2х-3)> 0 are called single-variable inequalities.

In general, this concept is defined as follows:

Definition. Let f(x) and g(x) be two expressions with variable x and domain X. Then an inequality of the form f(x) > g(x) or f(x)< g(х) называется неравенством с одной переменной. Мно­жество X называется областью его определения.

Variable value x from many x, under which the inequality turns into a true numerical inequality, is called its decision. Solving an inequality means finding the set of its solutions.

Thus, by solving inequality 2 x + 7 > 10 -x, x? R is the number x= 5, since 2 5 + 7 > 10 - 5 is a true numerical inequality. And the set of its solutions is the interval (1, ∞), which is found by performing the transformation of the inequality: 2 x + 7 > 10-x => 3x >3 => x >1.

Equivalent inequalities. Equivalence theorems for inequalities

The concept of equivalence underlies the solution of inequalities with one variable.

Definition. Two inequalities are said to be equivalent if their solution sets are equal.

For example, inequalities 2 x+ 7 > 10 and 2 x> 3 are equivalent, since their solution sets are equal and represent the interval (2/3, ∞).

Theorems on the equivalence of inequalities and their consequences are similar to the corresponding theorems on the equivalence of equations. When proving them, the properties of true numerical inequalities are used.

Theorem 3. Let the inequality f(x) > g(x) set on the set X and h(x) is an expression defined on the same set. Then the inequalities f(x) > g(x) and f(x) + h(x) > g(x) + h(x) are equivalent on the set x.

Consequences follow from this theorem, which are often used in solving inequalities:

1) If both sides of the inequality f(x) > g(x) add the same number d, then we get the inequality f(x) + d > g(x) + d, equivalent to the original.

2) If any term (a numerical expression or an expression with a variable) is transferred from one part of the inequality to another, changing the sign of the term to the opposite, then we obtain an inequality equivalent to the given one.

Theorem 4. Let the inequality f(x) > g(x) set on the set X and h(X X from many X expression h(x) takes positive values. Then the inequalities f(x) > g(x) and f(x) h(x) > g(x) h(x) are equivalent on the set x.

f(x) > g(x) multiply by the same positive number d, then we get the inequality f(x) d > g(x) d, equivalent to this one.

Theorem 5. Let the inequality f(x) > g(x) set on the set X and h(X) is an expression defined on the same set, and for all X their multitude X expression h(X) takes negative values. Then the inequalities f(x) > g(x) and f(x) h(x) > g(x) h(x) are equivalent on the set X.

The corollary follows from this theorem: if both sides of the inequality f(x) > g(x) multiply by the same negative number d and reverse the inequality sign, we get the inequality f(x) d > g(x) d, equivalent to this one.

Solving inequalities with one variable

Solve inequality 5 X - 5 < 2х - 16, X? R, and justify all the transformations that we will perform in the solution process.

Inequality solution X < 7 является промежуток (-∞, 7) и, сле­довательно, множеством решений неравенства 5X - 5 < 2x + 16 is the interval (-∞, 7).

Exercises

1. Determine which of the following entries are single-variable inequalities:

a) -12 - 7 X< 3x+ 8; d) 12 x + 3(X- 2);

b) 15( x+ 2)>4; e) 17-12 8;

c) 17-(13 + 8)< 14-9; е) 2x 2+ 3x-4> 0.

2. Is the number 3 a solution to the inequality 6(2x + 7) < 15(X + 2), X? R? And the number 4.25?

3. Are the following pairs of inequalities equivalent on the set of real numbers:

a) -17 X< -51 и X > 3;

b) (3 x-1)/4 >0 and 3 X-1>0;

c) 6-5 x>-4 and X<2?

4. Which of the following statements are true:

a) -7 X < -28 => x>4;

b) x < 6 => x < 5;

in) X< 6 => X< 20?

5. Solve inequality 3( x - 2) - 4(X + 1) < 2(х - 3) - 2 and justify all the transformations that you will perform in this case.

6. Prove that the solution of the inequality 2(x+ 1) + 5 > 3 - (1 - 2X) is any real number.

7. Prove that there is no real number that would be a solution to the inequality 3(2 - X) - 2 > 5 - 3X.

8. One side of the triangle is 5 cm, and the other is 8 cm. What can be the length of the third side if the perimeter of the triangle is:

a) less than 22 cm;

b) more than 17 cm?

GRAPHIC SOLUTION OF INEQUALITIES WITH ONE VARIABLE. For a graphical solution of the inequality f(x) > g(x) you need to plot function graphs

y = f(x) = g(x) and choose those intervals of the abscissa axis, on which the graph of the function y = f(x) located above the graph of the function y \u003d g(x).

Example 17.8. Solve Graphically an Inequality x 2- 4 > 3X.

Solution. Let us construct graphs of functions in one coordinate system

y \u003d x 2 - 4 and y= Zx (Fig. 17.5). It can be seen from the figure that the graphs of the functions at= x 2- 4 is located above the graph of the function y \u003d 3 X at X< -1 and x > 4, i.e. the set of solutions to the original inequality is the set

(- ¥; -1) È (4; + oo) .

Answer: x O(-oo; -1) and ( 4; +oo).

Graph of a quadratic function at= ax 2 + bx + c is a parabola with branches pointing upwards if a > 0, and down if a< 0. In this case, three cases are possible: the parabola intersects the axis Oh(i.e. the equation ah 2+ bx+ c = 0 has two different roots); the parabola touches the axis X(i.e. the equation ax 2 + bx+ c = 0 has one root); the parabola does not intersect the axis Oh(i.e. the equation ah 2+ bx+ c = 0 has no roots). Thus, there are six possible positions of the parabola, which serves as a graph of the function y \u003d ah 2+b x + c(Fig. 17.6). Using these illustrations, one can solve quadratic inequalities.

Example 17.9. Solve the inequality: a) 2 x r+ 5x - 3 > 0; b) -Zx 2 - 2x- 6 < 0.

Solution, a) The equation 2x 2 + 5x -3 \u003d 0 has two roots: x, \u003d -3, x 2 = 0.5. Parabola serving as a graph of a function at= 2x 2+ 5x -3, shown in fig. a. Inequality 2x 2+ 5x -3 > 0 is performed for those values X, for which the points of the parabola lie above the axis Oh: it will be at X< х х or when X> x r> those. at X< -3 or at x > 0.5. Hence, the set of solutions to the original inequality is the set (- ¥; -3) and (0.5; + ¥).

b) Equation -Zx 2 + 2x- 6 = 0 has no real roots. Parabola serving as a graph of a function at= - 3x 2 - 2x - 6 is shown in fig. 17.6 Inequality -3x 2 - 2x - 6 < О выполняется при тех значениях X, for which the points of the parabola lie below the axis Oh. Since the entire parabola lies below the axis Oh, then the set of solutions to the original inequality is the set R .

INEQUALITIES CONTAINING A VARIABLE UNDER THE MODULUS SIGN. When solving these inequalities, keep in mind that:

|f(x) | =

f(x), if f(x) ³ 0,

- f(x), if f(x) < 0,

In this case, the region of admissible values ​​of the inequality should be divided into intervals, on each of which the expressions under the modulus sign retain their sign. Then, expanding the modules (taking into account the signs of the expressions), you need to solve the inequality on each interval and combine the resulting solutions into a set of solutions to the original inequality.

Example 17.10. Solve the inequality:

|x -1| + |2-x| > 3+x.

Solution. The points x = 1 and x = 2 divide the real axis (ODZ of inequality (17.9) into three intervals: x< 1, 1 £ х £.2, х >2. Let's solve this inequality on each of them. If x< 1, то х - 1 < 0 и 2 – х >0; so |x -1| = - (x - I), |2 - x | = 2 - x. Hence, inequality (17.9) takes the form: 1- x + 2 - x > 3 + x, i.e. X< 0. Таким образом, в этом случае решениями неравенства (17.9) являются все отрицательные числа.

If 1 £ x £.2, then x - 1 ³ 0 and 2 - x ³ 0; therefore | x-1| = x - 1, |2 - x| = 2 - x. .So, there is a system:

x - 1 + 2 - x > 3 + x,

The resulting system of inequalities has no solutions. Therefore, on the interval [ 1; 2], the set of solutions to inequality (17.9) is empty.

If x > 2, then x - 1 > 0 and 2 - x<0; поэтому | х - 1| = х- 1, |2-х| = -(2- х). Значит, имеет место система:

x -1 + x - 2 > 3 + x,

x > 6 or

Combining the solutions found on all parts of the ODZ of inequality (17.9), we obtain its solution - the set (-¥; 0) È (6; + oo).

Sometimes it is useful to use the geometric interpretation of the modulus of a real number, according to which | a | means the distance of the point a of the coordinate line from the origin O, and | a - b | means the distance between points a and b on the coordinate line. Alternatively, you can use the method of squaring both sides of the inequality.

Theorem 17.5. If expressions f(x) and g(x) for any x take only non-negative values, then the inequalities f(x) > g(x) and f (x) ² > g (x) ² are equivalent.

58. Main conclusions § 12

In this section, we have defined the following concepts:

Numeric expression;

The value of a numeric expression;

An expression that doesn't make sense;

Expression with variable(s);

Expression scope;

identically equal expressions;

Identity;

Identity transformation of an expression;

Numerical equality;

Numerical inequality;

Equation with one variable;

Root of the equation;

What does it mean to solve an equation;

Equivalent Equations;

Inequality with one variable;

Solution of inequality;

What does it mean to solve an inequality;

Equivalent inequalities.

In addition, we considered theorems on the equivalence of equations and inequalities, which are the basis for their solution.

Knowledge of the definitions of all the above concepts and theorems on the equivalence of equations and inequalities is a necessary condition for a methodically competent study of algebraic material with younger students.